2002 Fiscal Year Final Research Report Summary
Finite Simple Groups and Related Codes, Lattices and Vertex Operator Algebras
| Project/Area Number |
12440003
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Chiba University |
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Principal Investigator |
KITAZUME Masaaki Chiba University, Faculty of Science, Professor, 理学部, 教授 (60204898)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIYAMA Ken-ichi Chiba University, Faculty of Science, Associate Professor, 理学部, 助教授 (90206441)
NOZAWA Sohei Chiba University, Faculty of Science, Professor, 理学部, 教授 (20092083)
KOSHITANI Shigeo Chiba University, Faculty of Science, Professor, 理学部, 教授 (30125926)
HARADA Masaaki Yamagata University, Faculty of Mathematical Sciences, Associate Professor, 理学部, 助教授 (90292408)
YAMADA Niromichi Hitotsubashi University, Graduate School of Economics, Professor, 経済学研究科, 教授 (50134888)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Finite Group / Simple Group / Sporadic Simple Group / Monster Simple Group / Code / Lattice / Design / Vertex Operator Algebra |
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| Research Abstract |
We have studied codes, lattices and vertex operator algebras related to finite simple groups. Main results are as follows :
1.The classifications of the 2-and 3-radical subgroups of Fischer's simple groups F_<22>, F_<23>, F'_<24> have been completed.
2.Even unimodular Gaussian lattices of dimension 12 have been classified.
3.A sufficient condition for extremal Z_6-codes, and as an application, many Z_6-code related the Leech lattice had been constructed. Moreover the self-dual Z_6-codes of length 8 are classified.
4.The construction method of unimodular lattices using ternary self-dual codes has been studied. Extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time.
5.The constructions of vertex operator algebras using Z_8-codes and subalgebras V_<√<2>A_3> have been considered. We also give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_3)^8 into the Leech lattice.
6.By using Z_3-orbifold construction given by Dong-Mason, we give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_2)^<12> the Leech lattice. We also give the explicit actions of certain 3A-elements of the Monster simple group on V^* are defined.
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